Transaction costs

In this section we investigate how transaction costs change our results from the previous sections. For simplicity we assume that the cost of a transaction is given by some fraction of the transaction size, i.e. we are only concerned with proportional transaction costs.

Note that the introduction of a fixed component in the transaction costs can result in the cushion becoming negative and thus lead to default risk if the cushion size is very small.

For economic reasons, in such cases, one would resort to changing the strategy and omit the transactions if the cushion size is very small. This problem can be avoided if only proportional transaction costs are considered.

For the definition of the transaction costs we follow Black and Perold (1992). Denote the proportional factor of the transaction costs by β such that e.g. β = 1% means that for any transaction,1%of the transaction size is lost in value. Further denote byC_τi andC_τi+

the cushion before and after the transaction, respectively, such that C_t+ is the cushion process net of transaction costs. We do not use the specific notation C^tr andC^Cap as the general procedure of the implementation of transaction costs holds for both strategies.

The trading rule of our strategy will be to invest the quantity mC_τi+ into the risky asset at time τ_i such that the transaction costs are immediately implemented in the strategy.

At time τ_i−1 the investment in the risky asset is mC_τi−1+ according to the trading rule.

At time τ_i this investment will have evolved to mC_τi−1+S^Sτi

τi−1 and the trading rule will require to invest the amount mC_τi into the risky asset. Therefore the transaction costs will be βm:::C_τi+−C_τi−1+ ^Sτi

S_τi−1

:::. We can now find the cushion net of transaction costs at time τ_i through the equation

C_τi+ =C_τiβm::

::C_τi+−C_τi−1+ S_τi S_τi−1

:::: (2.19)

and from the definition of the trading dates in (2.2) we know

C_τi =C_τi−1+e^{r(τi−τi−1)}k_u (2.20) and

C_τi =C_τi−1+e^{r(τi−τi−1)}k_d (2.21) for and up- and down-move respectively. Furthermore, a combination of equations (2.1) and (2.20) yields

S_τi

S_τi−1 = k_u+m−1

m e^{r(τi−τi−1)} (2.22)

in case of an up-move while a combination of equations (2.1) and (2.21) yields S_τi

S_τi−1 = k_d+m−1

m e^{r(τi−τi−1)} (2.23)

in case of a down-move. Combining equations (2.19), (2.20) and (2.22) now gives C_τi+ = C_τi−1+e^{r(τi−τi−1)}k_u−βm

C_τi+−C_τi−1+k_u+m−1

m e^{r(τi−τi−1)}

⇔ C_τi+ = C_τi−1+e^{r(τi−τi−1)}

1 + 1 +β

1 +βm(k_u−1) ( )* +

=:ˆku

(2.24)

in case of an up-move and likewise equations (2.19), (2.21) and (2.23) yield C_τi+ = C_τi−1+e^{r(τi−τi−1)}k_d+βm

C_τi+−C_τi−1+k_d+m−1

m e^{r(τi−τi−1)}

⇔ C_τi+ = C_τi−1+e^{r(τi−τi−1)}

1− 1−β

1−βm(1−k_d) ( )* +

=:ˆkd

(2.25)

in case of a down-move. It is easy to see that generally kˆ_u < k_u and ˆk_d < k_d for β > 0 such that a comparison with (2.2) leads to the insight that the cushion process net of transaction costs is gernerally smaller than the cushion process without transaction costs, as it should be. Note, that the cushion process is not supposed to become negative.

Therefore from (2.25) we find a lower bound for k_d to be given by kˆ_d ≥0 ⇔ k_d ≥1− 1−βm

1−β .

For 0< k_d <1− ^1−βm_1−β the cushion process without transaction costs will still always be positive but the transaction costs will cause the cushion to become negative on the first down-move such that the amount Gcan not be guaranteed any more.

It is important to keep in mind, that while the probability of an up-move or down-move of the cushion remains unchanged in the presence of transaction costs, i.e. still hinges on the triggers k_u and k_d, the discounted cushion only multiplies with ˆk_u < k_u in case of an up-move and ˆk_d< k_din case of a down-move. It is fairly easy to include transaction costs in the propositions of section 2.2. Basically, replacing k_u byˆk_u andk_dbyˆk_dwherever they don’t refer to a probability is all there is to do. For example, in proposition 2.2.3, k_u and k_d must be replaced in the expressions for n_x, y₁, y₂ while they must not be replaced in the expressions fora,band hence also the expressions foru,d,ρand qremain unchanged.

However, the condition k_d = _k¹

u must be changed to kˆ_d = _ˆ¹

ku. Generally, apart from the trivial case m = 1, if k_d = k⁻_u¹ holds, ˆk_d = ˆk⁻_u¹ will not hold. This condition can be satisfied by first calculating kˆ_u as defined in equation (2.24), then putting ˆk_d = _ˆ¹

ku and finally calculating k_d from kˆ_d, using the definition of kˆ_d in equation (2.25). It is slightly more difficult to include transaction costs in the propositions of section 2.3, in particular for the case mC0 < V0. The reason for this is the change in the trading rule at the first time the cap becomes binding. Since the cap is binding at that time, the investment into the risky asset is less than it would be without the cap and therefore the transaction costs are also less. In section 2.3 we have omitted to present the density of the terminal value of the capped CPPI. As an example of how to implement transaction costs, we will now give this density in the presence of transaction costs. The density without transaction costs can be deduced by setting β = 0.

Corollary 2.4.1 (Density of the capped CPPI, case mC₀ < V₀ +Z)

Let β ≥ 0 the factor for the transaction costs, k_u > 0, ˆk_u as in eq. (2.24), ˆk_d = _ˆ¹

ku, k_d= 1− ₁¹_−βm^−β (1−kˆ_d) and Z ∈R+ the maximum amount of borrowing allowed. Further let n¯ as in equation (2.11) and n_x, y₁(x), y₂(x) as in proposition 2.2.3 andC, n_x, y₁(x), y₂(x)as in proposition 2.3.1 withk_u andk_dexchanged bykˆ_u andˆk_d. Additionally letV :=

F0+C0ˆkⁿu^¯ku+Z+βC0ˆkⁿu^¯(ku+m−1)

1+β , a := ¹_σlog ^mC_V , y₃(x) := _σ¹ log^xe−rT_V ^+Z for all x ∈ [G,∞).

Then the density of the terminal value of the capped CPPI in the presence of transaction costs is given by:

p_VCap

T (x) =p₁(x) +p₂(x) where

p₁(x) =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ L⁻_s,T¹

qn¯(n_x, s)ρ(s, y1(x))^{∂ y}_{∂ x}¹ +qn¯(n_x+ 1, s)ρ(s, y2(x))^{∂ y}_{∂ x}²

, n_x <n¯ L⁻_s,T¹

q_n¯(n_x, s)ρ(s, y₁(x))^{∂ y1}

∂ x

, n_x = ¯n

0 , n_x >n¯

and

p₂(x) =

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩ L⁻_s,T¹

h(¯n+ 1, s)d(s)

q₀(n_x, s)ρ(s, y₁(x))^{∂ y1}

∂ x

+q0(n_x+ 1, s)ρ(s, y₂(x))^{∂ y}_{∂ x}²

, n_x <−1 L⁻_s,T¹

h(¯n+ 1, s)d(s)

q₀(n_x, s)ρ(s, y₁(x))^{∂ y}_{∂ x}¹

+q₀(n_x+ 1, s)ρ(s, y₂(x)|a,∞, δ)^{∂ y2}

∂ x

, n_x =−1 L⁻_s,T¹

h(¯n+ 1, s)d(s)q0(n_x, s)ρ(s, y₁(x)|a,∞, δ)^{∂ y}_{∂ x}¹ +h(¯n+ 1, s)ρ(s, y₃(x)|a,∞, δ)^{∂ y}_{∂ x}³

, n_x = 0 Proof: The expression forp₁(x)follows immediately from proposition 2.3.2 by differen-tiation. Forp₂(x) we have to take into account that after net n¯+ 1 up-moves (which we suppose to happen at time τ) the trading rule changes such that instead of mC_τ^Cap₊ the amount V_τ^Cap₊ +Ze^rτ is invested into the risky asset. Since mC_τ^Cap ≥ V_τ^Cap+Ze^rτ, the transaction costs for the changes made to the portfolio will therefore be lower. Denote the trading date before τ with τ, then the amount invested into the risky asset at time τ was mC_τ^Cap+ and it follows from equation (2.3) that this amount has evolved to

mC_τ^Cap+

S_τ

S_τ = C_τ^Cap+e^r(τ−τ)(k_u+m−1)

= C₀e^rτˆkⁿ_u^¯(k_u+m−1) up to time τ. It follows that

V e^rτ =V_τ^Cap+Ze^rτ−β

V e^rτ −C₀e^rτˆkⁿ_u^¯(k_u+m−1)

determines the amount V e^rτ to be invested into the risky asset at time τ. This equation can be solved for V to yield

V = V_τe^−rτ +Z+βC₀ˆk_u^¯n(k_u+m−1) 1 +β

0 5 10 15 20 25 30 Expected Number of Trading Dates 1150

1155 1160 1165 1170

ExpectedReturn

Figure 2.12: Expected terminal value of the sim-ple CPPI for different values of the transaction costs. The parameters are V₀ = 1000, G = 800, m = 4, μ = 0.15, r = 0.05, σ = 0.20, T = 1, Z = 0. From top to bottom the curves are for β = 0,β = 0.2%andβ= 0.5%.

0 5 10 15 20 25 30

Expected Number of Trading Dates 1142

1144 1146 1148

ExpectedReturn

Figure 2.13: Expected terminal value of the capped CPPI for different values of the transaction costs. The parameters are V₀ = 1000, G = 800, m = 4, μ = 0.15, r = 0.05, σ = 0.20, T = 1, Z = 0. From top to bottom the curves are for β = 0,β= 0.2%andβ= 0.5%.

from which it follows immediately that V matches the definition in the corollary if it is taken into account that

V_τ^Cap =e^rτ

F₀+C₀ˆk_u^¯nk_u

is the portfolio value at time τ before the transaction. Now, analogously to the proof of proposition 2.3.2, the expressions for a andy₃ can be found and the expression forp₂(x)

then follows from proposition 2.3.2 by differentiation. 2

Figure 2.12 shows the expected terminal value of the simple CPPI depending on the expected number of trading dates with and without transaction costs. While it can be shown that the expected terminal value of the simple CPPI in continuous time is greater than the expected terminal value of the simple CPPI in discrete time, i.e. the global maximum of the expected terminal value is attained for E[N] → ∞ or equivalently k_u → 1, this is not true any more if transaction costs are considered. In the presence of transaction costs, the expected terminal value exhibits a local maximum in the number of trading dates. While it is rather intuitive that a large number of trading dates causes the expected terminal value to decrease, it is surprising how few trading dates are sufficient to produce this effect. Figure 2.12 shows that the maximum of the expected terminal value is approximately at 7.5 expected trading dates for β = 0.2% and at 3.5 expected trading dates (per year!) for β = 0.5%. Note also that it is possible for the expected terminal value to have a local minimum for largek_u (or equivalently for very few expected trading dates). A similar effect occurs for m → ∞. Black and Perold (1992) show that the expected terminal value of the simple CPPI has a maximum for very large m before

0 5 10 15 20 T

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85

Probability

Z0 ZV0

Z

Figure 2.14: Probability of the CPPI and the capped CPPI performing better than the riskless asset. The parameters areV₀ = 1000, F₀ = 750, m = 4, μ = 0.15, r = 0.05, σ= 0.20, k_u = 1.01, Z=∞, V₀, 0.

0 5 10 15 20

T 0.2

0.3 0.4 0.5

Probability

Z0 ZV0

Z

Figure 2.15: Probability of the CPPI and the capped CPPI performing better than the riskless asset. The parameters areV₀ = 1000, F₀ = 750, m= 4, μ = 0.15, r = 0.05, σ = 0.30, k_u = 1.01, Z=∞, V₀, 0.

converging to the expected terminal value of a stop-loss strategy. Similarly, for large k_u the simple CPPI converges to a stop-loss strategy. With the help of the expected terminal value of a stop-loss strategy it can be shown, that in spite of the local minimum for large k_u, the global maximum can not be at k_u → ∞.

For the capped CPPI, the situation is different. The global maximum without transaction costs can be attained for E[N] → ∞ with the same effects as for the simple CPPI.

However, figure 2.13 shows, that the global maximum of the expected terminal value can also be attained for k_u → ∞ which reflects the case of a stop-loss strategy. This effect occurs if the initial exposure is close to or even greater than the maximum exposure. For initial exposures well below the maximum exposure, the situation will be as for the simple CPPI in figure 2.12.

Im Dokument CPPI Strategies in Discrete Time (Seite 87-92)